by Mehak
PRODUCTION FUNCTION
The production
function relates the output of a firm to the amount of inputs, typically
capital and labor.
It is important to
keep in mind that the production function describes technology, not economic
behavior. A firm may maximize its profits given its production function,
but generally takes the production function as a given element of that
problem. (In specialized long-run models, the firm may choose its capital
investments to choose among production technologies.)
CONCEPT
In micro-economics,
a production function is a function that specifies the output of a firm for all
combinations of inputs. A meta-production function (sometimes
metaproduction function) compares the practice of the existing entities
converting inputs into output to determine the most efficient practice
production function of the existing entities, whether the most efficient
feasible practice production or the most efficient actual practice production.[3]clarification needed In
either case, the maximum output of a technologically-determined production
process is a mathematical function of one or more inputs. Put another way, given the set
of all technically feasible combinations of output and inputs, only the
combinations encompassing a maximum output for a specified set of inputs would
constitute the production function. Alternatively, a production function can be
defined as the specification of the minimum input requirements needed to
produce designated quantities of output, given available technology. It is
usually presumed that unique production functions can be constructed for every
production technology.
By assuming that
the maximum output technologically possible from a given set of inputs is
achieved, economists using a production function in analysis are abstracting
from the engineering and managerial problems inherently associated with a
particular production process. The engineering and managerial problems of technical efficiency are assumed to be solved, so that analysis can focus
on the problems of allocative efficiency. The firm is assumed to be making allocative choices
concerning how much of each input factor to use and how much output to produce,
given the cost (purchase price) of each factor, the selling price of the
output, and the technological determinants represented by the production
function. A decision frame in which one or more inputs are held constant may be
used; for example, (physical) capital may be assumed to be
fixed (constant) in the short run,
and labour and possibly other inputs such as raw materials variable, while in
the long run, the quantities of
both capital and the other factors that may be chosen by the firm are variable.
In the long run, the firm may even have a choice of technologies, represented
by various possible production functions.
The relationship
of output to inputs is non-monetary; that is, a production function relates
physical inputs to physical outputs, and prices and costs are not reflected in
the function. But the production function is not a full model of the production
process: it deliberately abstracts from inherent aspects of physical production
processes that some would argue are essential, including error, entropy or
waste. Moreover, production functions do not ordinarily model the business processes, either,
ignoring the role of management. (For a primer on the fundamental elements of
microeconomic production theory, see production theory basics).
The primary
purpose of the production function is to address allocative efficiency in the
use of factor inputs in production and the resulting distribution of income to
those factors. Under certain assumptions, the production function can be used
to derive a marginal product for each factor, which implies an ideal division of
the income generated from output into an income due to each input factor of
production.
Specifying
the production function
A production function can be expressed in a functional form
as the right side of
where:
quantity of output
quantities of factor inputs
(such as capital, labour, land or raw materials).
If Q is
not a matrix (i.e. a scalar, a vector, or even a diagonal matrix), then this form does not
encompass joint production, which is a production process that has multiple
co-products. On the other hand, if f maps
from Rn to Rk then it is a joint production
function expressing the determination of k different types of output based on the joint usage of the
specified quantities of the ninputs.
One formulation, unlikely to be relevant in practice, is as
a linear function:
where
and
are parameters that are determined empirically.
The Leontief
production function applies
to situations in which inputs must be used in fixed proportions; starting from
those proportions, if usage of one input is increased without another being
increased, output will not change. This production function is given by
Other forms include the constant elasticity of substitution production function (CES), which is a generalized form
of the Cobb-Douglas function, and the quadratic production function. The best
form of the equation to use and the values of the parameters (
) vary from company to company and industry to industry. In a
short run production function at least one of the
's (inputs) is fixed. In the long run all factor inputs are
variable at the discretion of management.
Production
function as a graph
Quadratic Production Function
Any of these equations can be plotted on a graph. A typical
(quadratic) production function is shown in the following diagram under the
assumption of a single variable input (or fixed ratios of inputs so the can be
treated as a single variable). All points above the production function are
unobtainable with current technology, all points below are technically
feasible, and all points on the function show the maximum quantity of output
obtainable at the specified level of usage of the input. From the origin,
through points A, B, and C, the production function is rising, indicating that
as additional units of inputs are used, the quantity of output also increases.
Beyond point C, the employment of additional units of inputs produces no
additional output (in fact, total output starts to decline); the variable input
is being used too intensively. With too much variable input use relative to the
available fixed inputs, the company is experiencing negative marginal returns
to variable inputs, and diminishing total returns. In the diagram this is
illustrated by the negative marginal physical product curve (MPP) beyond point
Z, and the declining production function beyond point C.
From the origin to point A, the firm is experiencing
increasing returns to variable inputs: As additional inputs are employed,
output increases at an increasing rate. Both marginal
physical product (MPP,
the derivative of the production function) and average physical product (APP,
the ratio of output to the variable input) are rising. The inflection point A
defines the point beyond which there are diminishing marginal returns, as can
be seen from the declining MPP curve beyond point X. From point A to point C,
the firm is experiencing positive but decreasing marginal returns to the
variable input. As additional units of the input are employed, output increases
but at a decreasing rate. Point B is the point beyond which there are
diminishing average returns, as shown by the declining slope of the average
physical product curve (APP) beyond point Y. Point B is just tangent to the
steepest ray from the origin hence the average physical product is at a
maximum. Beyond point B, mathematical necessity requires that the marginal
curve must be below the average curve.
FOR THE ILLUSTRATIONOF THE PRODUCTION FUNCTION WE HAVE
TO GO THROUGH TO THE CONCEPT OF FIRM; BASIC CONCEPT RELATED TO A FIRM:
Objectives of
Firm
Like the rational
consumers aim at maximising their satisfaction or utility, the firms aim at
maximising their profits. Apart from profit maximisation, firms may aim at
sales maximisation, revenue maximisation, good will among the consumers.
Depending upon the type of ownership of a firm, the nature of objectives may
change. For example, it is argued that, under corporation as a form of firm’s
ownership, the objective of profit maximisation is replaced by the objective of
sales maximisation. This is because, in big corporations, ownership of firm is
separated from its management.
With this
background information about the firm, its ownership structure and its
objectives, let us begin with the analysis of the concept of production. A
story of production and firm’s behaviour will be easier to follow once we take
a note of following concepts:-
1.Prodution process
It is process by
which the inputs or factors of production are transformed into output. In a
cement factory, inputs include labour of its workers, raw materials such as
limestone, sand, clay, and capital invested in equipment required to produce
cement. Output of cement industry would be different varieties of cement.
2.Inputs or factors of production
There are four
factors of production, land, labour, capital and organisation. All these are
brought together in the process of production to form a final output. Land
represents natural resources like land plots, minerals, water, oil, etc. Labour
is considered to be an integral part of the process of production. Both skilled
and unskilled labour is required by the firm. Capital represents physical
capital in the form of machinery, equipment, plants, factory and other physical
assets. Finally, organisation/entrepreneur brings all these factors of
production together to transform them into a finished product.
3.Short run and the long run period
In the theory of
production, short run is a period during which some of the factors of
production mentioned above are constant. For example, in the short run, firm
can not buy a new machine. So capital may remain constant in the short run. If
it has to increase production in the short run, it may do so by hiring more
contract labour to work on the same stock of machines or equipment. Long run,
on the other hand, is a period, during which all the factors of production can
vary. A firm can not only hire more/less labour but also can increase/reduce
size of plant, buy more/sale existing stock of capital, and so on. One should
keep in mind, the short-run and long-run period in production theory, is not
time specific. For a poultry firm, for example, long run will be a period,
till it increases its capacity by adding poultry stock (which may take say 2
weeks). But for a cement factory, it may take 2 years to increase its capacity
by constructing a new plant. So long run for cement factory may be 2 years.
TYPES OF PRODUCTION FUNCTION
There are two
distinct types of production function that show possible range of substitution
inputs in the production process.
1. Fixed
proportion Production function
2. Variable
proportions production function
These two types
are based on the technical coefficient of production. The technical
co-efficient is the amount of input required to produce a unit of output. For
example, if 50 workers are required to produce 200 units of output, then 0.25
is the technical co-efficient of labour for production.
When 0.25 units of
labour are required to produce every unit of output, it is called fixed
proportion production function. Here, doubling of quantities of capital and
labour in a required ratio will double the output. Fixed proportion production
function can be illustrated with the help of isoquants. In this type of
production function, the two factors of production, say labour and capital,
should be used in a fixed proportion. The isoquants of such function are right
angled as shown in the following diagram.
On the other hand,
when the technical co-efficient to produce different units of output is varying
or changing, it is called as the variable proportions production function. In
such a type of production function, given amount of output can be produced with
several alternative compbinations of labour and capital. Many commodities in
real world are produced with variable proportion production function. For
example, certain amount of wheat may be produced using more labour and less
capital in India and more capital and less labour in USA. Variable proportion
production function is illustrated in the following diagram.
